Derived categories of Quot schemes on smooth curves and tautological bundles

TL;DR

This paper introduces a categorical action of a quantum loop group on derived categories of Quot schemes on smooth curves, leading to a semi-orthogonal decomposition and cohomology calculations of tautological bundles.

Contribution

It establishes a new categorical framework linking quantum groups and Quot schemes, enabling decomposition and cohomology computations.

Findings

Semi-orthogonal decomposition of Quot scheme derived categories

Calculation of tautological bundle cohomology

Connection between quantum groups and geometric categories

Abstract

We define a categorical action of the shifted quantum loop group of $\mathfrak{sl}_2$ on the derived categories of Quot schemes of finite length quotient sheaves on a smooth projective curve. As an application, we obtain a semi-orthogonal decomposition of the derived categories of Quot schemes, of representation theoretic origin. We use this decomposition to calculate the cohomology of interesting tautological vector bundles over the Quot scheme.